论文标题
均匀剪切流的玻尔兹曼方程
The Boltzmann equation for uniform shear flow
论文作者
论文摘要
稀有气体的均匀剪切流受到时间依赖性的空间均匀玻尔兹曼方程，并具有线性剪切力。这种流动的主要特征是，由于诱导粘性热的剪切运动，温度可能会在时间上升高，并且该系统远离平衡。对于麦克斯韦分子，我们为任何小的剪切速率建立了独特的存在，规律性，剪切率依赖性结构和自相似曲线的非阴性。即使在空间不均匀的扰动框架中，也可以通过巨大的渐近稳定性来证明非负性是合理的，并且收敛的指数速率也与与第二阶剪切速率成正比的尺寸获得。该分析支持了数值结果，即自相似的轮廓接受代数高速尾巴，这是在证明中克服的关键困难。
The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. The analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof.
